Laboratory work #8a(15)
Calculation ultimate compressing and ultimate tension force for a finned panel.
For a finned panel the following data are given: distance between ribs “a”, distance
between longitudinal reinforcements t, thickness of the skin s , quantity of stringers nst, area
of cross sections of spars Fsp, type rolling section of stringer (angle bar, channel bar, tee bar,
zee bar etc.) and number of a standard size within the limits of appropriate gage. The
quantity of spar caps is equal nsp=2. The cross section of the panel is shown in the fig. 1. In
figure the circles show stringers, rectangular – spar cap. The quantity of stringers nst is
accepted equal nst = 4 for definiteness of fig. 1.
t
t
t
t
t
Fig. 1. Cross section of a finned panel
Material of the skin and stringers is aluminum alloy type 2024 with such mechanical
characteristics E = 72 GPa, lp = 270 MPa, y = 300 MPa, ut = 440 MPa. A material of spar
caps is stainless steel type 4340 with such mechanical characteristics E = 210 GPa,
lp = 850 MPa, y = 900 MPa, ut = 1100 MPa.
For ultimate compressing force Puc the parity is fair:
с
Puc  nst ( fst  f stjs )   сst .u  2  (Fsp  f sp
js )   sp.u ,
(1)
where Fsp – is area of cross section for spar from initial date, fst - is area of cross section for
stringer from gage, f jsst , f jssp are areas of the joined skin (effective width) to a stringer and a
spar cap accordingly which you should calculate, сst .u , csp.u are ultimate stresses stringer
and spar cap at compression.
The joined skin (effective width) is understood conditional width of skin, which has
the same material, stress, deformation, as the longitudinal reinforcements, and transfers the
same effort, that the real skin with width t. Thickness and length real and joined skin
coincide.
Ultimate stresses of stringers and spar caps at compression are equal accordingly:
l
g
с
 сst .u  min(  cr
.st ,  cr .st ) ,  sp.u =  ut ,
(2)
1
g
l
where  cr
.st ,  cr .st - are critical stress of loss of stability stringers under the local and
general forms accordingly,  ut - stress of ultimate tension for a material of spar caps. The
g
l
calculation of stresses  cr
.st ,  cr .st is considered below.
In case of fastening of stringer with the help by single-row riveting you have:
st
j .s
f
 2c  s ,
(3)
where 2c – is the width of the joined skin.
As the materials of the skin and stringer are identical, the parity is applicable for 2c:
2c
 1.9   s 
st
Е s /  st .
(4)
where Es – is Young’s modulus for skin, σst – is the stress in stringer.
As to 2csp, it is usual neglect by size f
sp
j .s
.
l
At calculation  cr
.st the structure is conditionally dismembered on plates. In the fig. 2
such partitions are shown on examples angle bar and zee bar. The formulas are used:
l .E
E
 cr
.st  min {  cr . p , i } ;


l
cr . st ,i
l ,E
cr . st , i
  ut 

0.9  k i  E i
( bi / 
1 
1   
 
ut
/
2
i
)
(5)
1
,
2
,
l .E
cr . st ,i
(6)
(7)
.
(8)
where b i ,  i - are width and thickness i-th of a plate (fig. 2).
The attaching factors k  i are undertaken according to the fig. 3. Let's pay attention,
that the thickness  i for the party of a bar connected to the skin is equal to the sum of shelf
thickness of a bar and thickness of the skin.
g
At calculation  cr
.st the model is used shown in a fig. 4. Coordinate of a center of
gravity of compound section you should compute from ratio:
y cg 

 0.5  f jst.s   s  f st  y cg .st
f jsst  f
.
(9)
st
g
The calculation  cr
.st assumes iterative process by calculation 2 с. For initial iteration
it is necessary to take model from fig. 4.
2
2 c  1.9   s 
l
E s /  cr
.st .
b1
b1
1
2
 2 3
1
b2
b2
b3
Fig. 2. Partition of bar on plate
The known Euler’s formula is applied:

g .E
cr .st


x cg
st
f js 
2 E I
( a ) 2 (
f st )
,
(10)

- central moment of inertia of
where  - is the attaching factor; a – rib pitch; I cg
compound section. It is considered that the loss of stability is possible in direction,
perpendicular skin, in view of high rigidity by last in the own plane.
For real stringers it is possible to accept 2 = 0.5.
I

x .cg
 I
x st
 f
st

( y cg .st  y cg
) 2  f jsst.  s / 12  2 c  s ( y cg  0.5  s ) 2
2
(11)
where Ix st - is moment of inertia for stringer from gage.
g .E
By determined stress  cr
.st , with the help of the empirical formula (7) we
g
calculate  cr
.st . The size  is calculated as:
g .E
   ut /  cr
.st .
(12)
It is result of the first iteration, but we are limited to it.
Now meanings of all sizes are known which are included in expression (1) and you can
determine ultimate compressing force of the panel Puс .
For ultimate tension force of the panel Put you should use ratio:
Put  n st  ( k 2 st fst  k3   р  s  t )   tst .u  2  Fsp  k 2 sp  tsp.u ,
(13)
3
where t – is the distance between longitudinal reinforcements; k2.st, k3, k2.sp - statistical
factors which are taking into account presence at longitudinal reinforcements and the skin of
holes under rivets or screws, available round of skin, backlog of stress in the skin;  р - the
reduction factor which is taking into account distinction in the diagrams -, if the materials
of stringers and skin have the different mechanical characteristics.
Fig. 3. Experimental attaching factor.
2c
yycgцт
x
Fig. 4. Model for calculation of inertia moment.
In the manuals the following recommendations are given:
k 2 st = 0.9 ; k 3 = 0.7-0.8 ; k 2 sp = 0.9-0.95.
As the materials of the skin and stringers are identical in our laboratory work than
4
 р = 1.
For calculation ultimate tension stress the dependences are used:
 tst .u  k 1 st   ut .st ; tsp.u  k 1 sp   ut .sp .
(14)
In view of alignment of stress in process of growth of plastic deformations the factors к 1 st ,
к 1 sp , which are received experimentally, are close to unit. For an aluminum alloy type 2024
k1 st = 0.8 … 0.85, for stainless steel type 4340 k1 sp= 0.95.
Then the meanings of all sizes, which are included in expression (13), are known, that
allows determining ultimate compressing force of the panel Put .
Labwork8(15)ultforce.doc
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